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I am currently decoding a $[8,4,4]$ binary code using syndrome decoding, and I have two words I'd like to decode. Here is the parity-check matrix I am working with:

$H = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix} $

For the first word $y_1=(11010011)$, the decoding algorithm didn't work because I didn't find the right syndrome. Indeed, $Hy_1^T=(0110)$ which is not in the syndrome table of $H$. For the second word $y_2=(01010001)$, the decoding worked and gave me a code element $c=(11010001)$. After looking at $c$, I realized that the first word I received was simply $c-(00000010)$, in other words there was a simple error of weight $1$. Is it normal that the algorithm couldn't decode it? Did I make an error computing the syndromes?

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I get $Hy_1^T = (0010)$, I think you miscomputed.. Note that this is the 7th column of $H$ so the most likely error is $(00000010)$, which it indeed was.

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  • $\begingroup$ My bad, somehow three people were on this and we all got it wrong, wew. Thanks a lot! $\endgroup$
    – Saegusa
    Jun 5, 2021 at 13:51

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